An intermixed iteration for constrained convex minimization problem and split feasibility problem

R E S E A R C H

Open Access

An intermixed iteration for constrained

convex minimization problem and split

feasibility problem

Kanyanee Saechou

_{and Atid Kangtunyakarn}

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand

Abstract

In this paper, we first introduce the two-step intermixed iteration for finding the common solution of a constrained convex minimization problem, and also we prove a strong convergence theorem for the intermixed algorithm. By using our main theorem, we prove a strong convergence theorem for the split feasibility problem. Finally, we apply our main theorem for the numerical example.

MSC: 46N10; 47H09; 74G60

Keywords: Constrained convex minimization problem; Split feasibility problem; Variational inequality

1 Introduction

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH.

We denote the fixed point set of a mappingT byF(T). Fixed point theory can be ap-plied to variational inequality problems, equilibrium problems, split feasibility problems, optimization problems, etc. These problems are encountered in various fields such as en-gineering, physics, game theory, and economics.

A mappingTofCinto itself is callednonexpansiveif

Tx–Ty ≤ x–y, ∀x,y∈C.

In mathematics, conventional optimization problems arise in the process of making a trading system more effective and are usually stated in terms of minimization problems. In this paper, we give a new iteration for solving two constrained convex minimization problems.

Convex constrained minimization problem is popular and very important to various branches in physics, engineering and economics, e.g., to find the minimum travel dis-tance or to find the lowest cost. Consider the constrained convex minimization problem as follows:

minimizef(x) :x∈C, (1)

where f :C→Ris a real-valued convex function. If f is (Fréchet) differentiable, then the gradient-projection algorithm (GPA) generates a sequence {xn}using the following

recursive formula:

xn+1=PC

xn–λ∇f(xn)

, ∀n≥0, (2)

or more generally,

xn+1=PC

xn–λn∇f(xn)

, ∀n≥0, (3)

where both in (2) and (3) the initial guessx0is taken fromCarbitrarily, and the parameters,

λorλn, are positive real numbers satisfying certain conditions. The convergence of the

algorithms (2) and (3) depends on the behavior of the gradient∇f. In fact, it is known that if∇f isα-strongly monotone andL-Lipschitz with constantsα,L≥0, then the operator

T:=PC(I–λ∇f) (4)

is a contraction; hence, the sequence{xn}defined by the algorithm (2) converges in norm

to the unique minimizer of (1). However, if the gradient∇f fails to be strongly monotone, the operatorTdefined by (4) could fail to be contractive; consequently, the sequence{xn}

generated by the algorithm (2) may fail to converge strongly [1]. If∇f is Lipschitz, then the algorithms (2) and (3) can still converge in the weak topology under certain conditions [2–4].

The variational inequality problem is to find a pointu∈Csuch that

v–u,Au ≥0, ∀v∈C. (5)

We denote the set of solutions of the variational inequality byVI(C,A). Many models of variational inequalities are used in practice, including a mathematical theory, some inter-esting connections to numerous disciplines and a wide range of important applications in engineering, physics, optimization, minimax problems, game theory, and economics; for more details, see [5,6].

Su and Xu [3] introduced the relation of a solution to the minimization problem (1) and solutions of the variational inequality (5) as stated in the following Lemma1, and this lemma helps to prove the theorem about the minimization problem more effectively; for more details, see [7–9].

Lemma 1 (Optimality condition, [3]) A necessary condition for a point x∗∈C to be a solution of the minimization problem(1)is that x∗solves the variational inequality

∇fx∗,x–x∗≥0, ∀x∈C. (6)

Equivalently,x∗∈C solves the fixed point equation

x∗=PC

x∗–λ∇fx∗,

ByUf we denote the set of solutions of (1).

In 2011, Ceng et al. [10] introduced the following iterative scheme that generates a se-quence{xn}in an explicit way:

xn+1=PC

snγVxn+ (I–snμF)Tnxn , ∀n≥0,

wheresn= 2–λ₄nL andPC(I–λn∇f) =snI+ (1 –sn)Tnfor eachn≥0. He proved that the

sequence{xn}converges strongly to a minimizerx∗∈Sof (1).

In 2014, Ming and Lei [11] introduced an explicit composite iterative method for finding the common element of the set of solutions to an equilibrium problem and the solution set to a constrained convex minimization problem, as well as proved a strong convergence theorem, as follows:

Algorithm 1 Givenx1∈C, let the sequences{un}and{xn}be generated iteratively by ⎧

⎨ ⎩

φ(un,y) +β1ny–un,un–xn ≥0, ∀y∈C,

xn+1=αnγVun+ (I–αnA)Tnun, ∀n∈N,

whereTnis a nonexpansive mapping fromPC(I–λn∇f) =snI+ (1 –sn)Tn, which is2+λ₄nL

-averaged with sn= 2–λ₄nL, and∇f is anL-Lipschitz mapping, for allL≥0,V :C→C

is anl-Lipschitz mapping with constantl≥0,A:C→Cis a strongly positive bounded linear operator with coefficientγ¯≥0 and 0 <γ<γ¯_l,un=Qβnxn,{λn} ⊂(0,

2

L),{αn} ⊂(0, 1), {βn} ⊂(0,∞) and{sn} ⊂(0,1₂).

In 2015, Yao et al. [12] introduced the intermixed algorithm for two strict pseudocon-tractionsSandTas follows:

Algorithm 2 For arbitrarily givenx0∈C,y0∈C, let the sequences{xn}and{yn}be

gen-erated iteratively by

⎧ ⎨ ⎩

xn+1= (1 –βn)xn+βnPC[αnf(yn) + (1 –k–αn)xn+kTxn], n≥0,

yn+1= (1 –βn)yn+βnPC[αng(xn) + (1 –k–αn)yn+kSyn], n≥0,

(7)

whereS,T:C→Careλ-strictly pseudocontractions,f:C→His aρ1-contraction, and g:C→His aρ2-contraction,k∈(0, 1–λ) is a constant, and{αn},{βn}are two real number

sequences in (0, 1).

Furthermore, under some control conditions, they proved that the iterative sequences

{xn}and{yn}defined by (7) converge independently toPF(T)f(y∗) andPF(S)g(x∗),

respec-tively, wherex∗∈F(T) ={z∈C:Tz=z}andy∗∈F(S) ={z∗∈C:Tz∗=z∗}.

Motivated by Yao et al. [12] and Ming et al. [11], we introduce the new iterative method as follows:

Algorithm 3 Givenx1,y1∈C, let the sequences{xn}and{yn}be defined by ⎧

⎨ ⎩

xn+1= (1 –μn)xn+μnPC(αnf(yn) + (1 –αn)T f1 nxn),

yn+1= (1 –μn)yn+μnPC(αng(xn) + (1 –αn)T f2 nyn),

where f,g: H→H are af- and ag-contraction mappings withaf,ag ∈(0, 1) and a=

max{af,ag}, ∇fi is an _L1_i-inverse strongly monotone with Li ≥0, for alli= 1, 2, {μn},

{αn} ⊆[0, 1],PC(I–λin∇fi) =sinI+ (1 –sin)T fi

n,∀i= 1, 2 andsin= 2–λi_nLi

4 ,{λ i

n} ⊂(0,L2i) and 0 <θ≤μn≤θ for alln∈Nand for someθ,θ> 0.

The purpose of this article is to combine the GPA and averaged mapping approach to design a two-step intermixed iteration for finding the common solution of a constrained convex minimization problem, and also prove a strong convergence theorem for the in-termixed algorithm generated by (8). Applying our main result, we prove a strong conver-gence theorem for the split feasibility problem. Moreover, we utilize our main theorem in the numerical example.

2 Preliminaries

Throughout this article, we always assume thatCis a nonempty, closed, and convex subset of a real Hilbert spaceH. We use “ ” for weak convergence and “→” for strong conver-gence. For everyx∈H, there is a unique nearest pointPCxinCsuch that

x–PCx ≤ x–y, ∀y∈C.

Such an operatorPCis called the metric projection ofHontoC.

Assume thatCis a nonempty closed and convex subset ofH. A mappingV:C→Cis said to be anl-Lipschitz if there exists a constantl≥0 such that

Vx–Vy ≤lx–y, ∀x,y∈C.

Ifl∈[0, 1), thenVis called a contraction. Obviously, ifl= 1,Vis a nonexpansive mapping.

Definition 1 A mappingT:H→His said to be firmly nonexpansive if and only if 2T–I

is nonexpansive, or equivalently,

x–y,Tx–Ty ≥ Tx–Ty2, x,y∈H.

Alternatively,T is firmly nonexpansive if and only ifT can be expressed as

T=1 2(I+S),

whereS:H→His nonexpansive.

Definition 2(Positive operator) An operatorAis calledpositiveif it is self-adjoint and

Ax,x ≥0 for allx∈H.

An operatorAonHis strongly positive if there exists a constantγ > 0 with the property

Lemma 2([13]) For a given z∈H and u∈C,

u=PCz ⇐⇒ u–z,v–u ≥0, ∀v∈C.

Furthermore,PCis a firmly nonexpansive mapping of H onto C.

Lemma 3([14]) Let H be a real Hilbert space.Then the following results hold:

(i) For allx,y∈Handα∈[0, 1],

αx+ (1 –α)y2=αx2+ (1 –α)y2–α(1 –α)x–y2,

(ii) x+y2_{≤ x}2_{+ 2y,x+y,for eachx,y∈H.}

Lemma 4([4]) Let{sn}be a sequence of nonnegative real numbers satisfying

sn+1= (1 –αn)sn+δn, ∀n≥0,

where{αn}is a sequence in(0, 1)and{δn}is a sequence such that (1) ∞_n=1αn=∞,

(2) lim sup_n→∞δn αn ≤0or

_∞

n=1|δn|<∞. Thenlimn→∞sn= 0.

Definition 3 A mappingT:H→His said to be anaveraged mappingif it can be written as the average of the identityIand a nonexpansive mapping, that is,

T= (1 –α)I+αS, (9)

whereα is a number in (0, 1) andS:H→H is nonexpansive. More precisely, when (9) holds, we say thatT isα-averaged.

Clearly, a firmly nonexpansive mapping is a1₂-averaged mapping.

Proposition 1 For given operators S,T,V:H→H:

(i) IfT= (1 –α)S+αVfor someα∈(0, 1)and ifSis averaged andV is nonexpansive,

thenT is averaged.

(ii) Tis firmly nonexpansive if and only if the complementI–T is firmly nonexpansive. (iii) IfT= (1 –α)S+αVfor someα∈(0, 1),Sis firmly nonexpansive andVis

nonexpansive,thenT is averaged.

(iv) The composition of finitely many averaged mappings is averaged.That is,if each of the mappings{Ti}Ni=1is averaged,then so is the compositionT1◦T2◦ · · · ◦TN.In

particular,ifT1isα1-averaged,andT2isα2-averaged,whereα1,α2∈(0, 1),then the

compositionT1◦T2isα-averaged,whereα=α1+α2–α1α2.

Lemma 5([11]) For given x∈H and let PC:H→C be a metric projection.Then (a) z=PCxif and only ifx–z,y–z ≤0,∀y∈C.

(b) z=PCxif and only ifx–z2≤ x–y2–y–z2,∀y∈C. (c) PCx–PCy,x–y ≥ PCx–PCy2,∀x,y∈H.

Lemma 6 ([15]) Each Hilbert space H satisfies Opial’s condition, i.e.,for any sequence {un} ⊂H with un u,the inequality

lim inf

n→∞ un–u<lim infn→∞ un–v

holds for every v∈H with v=u.

Definition 4 A nonlinear operatorT whose domainD(T)⊆Hand rangeR(T)⊆His

said to be:

(a) monotone if

x–y,Tx–Ty ≥0, ∀x,y∈D(T);

(b) β-strongly monotone if there existsβ> 0such that

x–y,Tx–Ty ≥βx–y2, ∀x,y∈D(T);

(c) v-inverse strongly monotone (for short,v-ism) if there existsv> 0such that

x–y,Tx–Ty ≥vTx–Ty2, ∀x,y∈D(T).

Proposition 2 Let T be an operator from H to itself.Then (a) Tis nonexpansive if and only if the complementI–Tis 1

2-ism; (b) IfTisv-ism,then forγ> 0,γT is_γv-ism;

(c) Tis averaged if and only if the complementI–Tisv-ism for somev>1

2.Indeed,for

α∈(0, 1),Tisα-averaged if and only ifI–Tis ₂1_α-ism.

Lemma 7([16]) Assume A:H→H is a strongly positive bounded linear operator with

coefficientγ > 0and0 <t≤ A–1.ThenI–tA ≤1 –tγ.

3 Main results

LetV :C→Cbel-Lipschitz with coefficientl≥0, andA:C→Ca strongly positive bounded linear operator with coefficientγ and 0 <γ <γ_l. Letf :C→Rbe a real-valued convex function and assume that∇f is anL-Lipschitz mapping withL≥0. From Xu [1], we have thatPC(I–λ∇f) is2+₄λL-averaged for 0 <λ<2_Land for eachn∈N, that is, we can

write

PC(I–λn∇f) = (1 –sn)I+snTnf,

whereTnf is nonexpansive andsn=2+λ4nL.

Theorem 1 Let C be a nonempty closed convex subset of a real Hilbert space H.For every i= 1, 2,fi:C→Rbe a real-valued convex function and assume that∇fiis an _L1_i-inverse strongly monotone with Li> 0and Ufi=∅. Let f,g:H→H be af- and ag-contraction

be generated by x1,y1∈C and ⎧

⎨ ⎩

xn+1= (1 –μn)xn+μnPC(αnf(yn) + (1 –αn)T f1 nxn),

yn+1= (1 –μn)yn+μnPC(αng(xn) + (1 –αn)T f2 nyn),

(10)

where{μn},{αn} ⊆[0, 1],PC(I–λin∇fi) =sinI+ (1 –sin)T fi

n,sin= 2–λi

nLi 4 and{λ

i

n} ⊂(0,L2i)for

all i= 1, 2.Assume that the following conditions hold:

(i) limn→∞αn= 0and ∞

n=1αn=∞,

(ii) 0 <θ≤μn≤θfor alln∈Nand for someθ,θ> 0, (iii) ∞_n=1|αn+1–αn|<∞,

∞

n=1|μn+1–μn|<∞.

Then{xn}and{yn}converge strongly as sin→0 (⇐⇒λin→L2i)∀i= 1, 2,to x

∗_=P

Uf1f(y

∗₎

and y∗=PUf2g(x

∗_{),respectively.}

Proof First, we show that{xn}and{yn}are bounded. Assume thatx∈Uf1 andy∈Uf2.

Then we have

xn+1–x=(1 –μn)xn+μnPC

αnf(yn) + (1 –αn)T f1 nxn

–x

=(1 –μn)(xn–x) +μn

PC

αnf(yn) + (1 –αn)T f1 nxn

–x

≤(1 –μn)xn–x+μnαnf(yn) + (1 –αn)Tnf1xn–x ≤(1 –μn)xn–x+μn

αnf(yn) –x+ (1 –αn)Tnf1xn–x

≤(1 –μn)xn–x+μn

αnf(yn) –x+ (1 –αn)xn–x

= (1 –αnμn)xn–x+αnμnf(yn) –x

≤(1 –αnμn)xn–x+αnμnf(yn) –f(y)+f(y) –x

≤(1 –αnμn)xn–x+αnμnayn–y+αnμnf(y) –x. (11)

Similarly, we get

yn+1–y ≤(1 –αnμn)yn–y+αnμnaxn–x+αnμng(x) –y. (12)

Combining (11) and (12), we have

xn+1–x+yn+1–y ≤

1 –αnμn(1 –a)

xn–x+yn–y

+αnμnf(y) –x+g(x) –y.

By induction, we can derive that

xn–x+yn–y ≤max

x1–x+y1–y,f(y) –x+g(x) –y,

for everyn∈N. This implies that{xn}and{yn}are bounded.

Next, we show thatxn+1–xn →0 andyn+1–yn →0. Observe that

Tf1 nxn–T

f1 n–1xn–1

≤Tf1

≤ xn–xn–1+

4PC(I–λ1n∇f1) – (2 –λ1nL1)

2 +λ1 nL1

xn–1

–

4PC(I–λ1n–1∇f1) – (2 –λ1n–1L1)

2 +λ1_n–1L1

xn–1

≤ xn–xn–1+

4PC(I–λ1n∇f1)

2 +λ1 nL1

xn–1–

4PC(I–λ1n–1∇f1)

2 +λ1_n–1L1

xn–1

+

2 –λ1_n–1L1

2 +λ1_n–1L1

xn–1–

2 –λ1_nL1

2 +λ1 nL1

xn–1

=xn–xn–1

+4(2 +λ

1

n–1L1)PC(I–λ1n∇f1)xn–1– 4(2 +λ1nL1)PC(I–λ1n–1∇f1)xn–1

(2 +λ1

nL1)(2 +λ1n–1L1)

+(2 –λ

1

n–1L1)(2 +λ1nL1)xn–1– (2 –λ1nL1)(2 +λ1n–1L1)xn–1

(2 +λ1

nL1)(2 +λ1n–1L1)

=xn–xn–1

+4(2 +λ

1

n–1L1)PC(I–λ1n∇f1)xn–1– 4(2 +λ1nL1)PC(I–λ1n–1∇f1)xn–1

(2 +λ1

nL1)(2 +λ1n–1L1)

+

4L1|λ1n–λ1n–1|

(2 +λ1

nL1)(2 +λ1n–1L1)

xn–1

=xn–xn–1

+4L1(λ

1

n–1–λ1n)PC(I–λ1n∇f1)xn–1

(2 +λ1

nL1)(2 +λ1n–1L1)

+4(2 +λ

1

nL1)(PC(I–λ1n∇f1)xn–1–PC(I–λ1n–1∇f1)xn–1)

(2 +λ1

nL1)(2 +λ1n–1L1)

+

4L1|λ1n–λ1n–1|

(2 +λ1

nL1)(2 +λ1n–1L1)

xn–1

≤ xn–xn–1

+4L1|λ

1

n–1–λ1n|PC(I–λ1n∇f1)xn–1

(2 +λ1

nL1)(2 +λ1n–1L1)

+4λ

1

n–1∇f1xn–1–λ1n∇f1xn–1

2 +λ1_n–1L1

+

4L1|λ1n–λ1n–1|

(2 +λ1

nL1)(2 +λ1n–1L1)

xn–1

≤ xn–xn–1+L1λ1n–1–λ1nPC

I–λ1_n∇f1

xn–1

+ 4λ1_n–1–λ_n1∇f1xn–1+L1λ1n–λ1n–1xn–1

≤ xn–xn–1

+λ1_n–1–λ1_nL1PC

I–λ1_n∇f1

xn–1+ 4∇f1xn–1+L1xn–1

≤ xn–xn–1+M1λ1n–1–λ1n, (13)

From the definition ofxnand (13), we have

xn+1–xn

=(1 –μn)xn+μnPC

αnf(yn) + (1 –αn)Tnf1xn

–(1 –μn–1)xn–1+μn–1PC

αn–1f(yn–1) + (1 –αn–1)T f1

n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μnPC

αnf(yn) + (1 –αn)Tnf1xn

–PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μnαnf(yn) + (1 –αn)Tnf1xn–

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μnαnf(yn) –αn–1f(yn–1)+(1 –αn)Tnf1xn– (1 –αn–1)T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μn

αnf(yn) –f(yn–1)+|αn–αn–1|f(yn–1)

+ (1 –αn)Tnf1xn–T f1

n–1xn–1+|αn–1–αn|T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μn

αnf(yn) –f(yn–1)+|αn–αn–1|f(yn–1)

+ (1 –αn)

xn–xn–1+M1λ1n–1–λ1n+|αn–1–αn|T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μn)xn–xn–1+|μn–1–μn|xn–1

+μn

αnf(yn) –f(yn–1)+|αn–αn–1|f(yn–1)

+ (1 –αn)xn–xn–1+ (1 –αn)

4M1 L1

s1_n–s1_n–1

+|αn–1–αn|T f1 n–1xn–1

+|μn–μn–1|PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

≤(1 –μnαn)xn–xn–1+|μn–1–μn|xn–1

+|μn–μn–1|PC

+μn

αnayn–yn–1+|αn–αn–1|f(yn–1)

+ (1 –αn)

4M1 L1

s1_n–s1_n–1+|αn–1–αn|T f1 n–1xn–1

. (14)

Using the same method as derived in (14), we have

yn+1–yn

≤(1 –μnαn)yn–yn–1+|μn–1–μn|yn–1

+|μn–μn–1|PC

αn–1g(xn–1) + (1 –αn–1)T f2 n–1yn–1

+μn

αnaxn–xn–1+|αn–αn–1|g(xn–1)

+ (1 –αn)

4M2 L2

s2_n–s2_n–1+|αn–1–αn|T f2 n–1yn–1

, (15)

for someM2> 0 such thatM2≥L2PC(I–λ2n∇f2)yn–1+ 4∇f2yn–1+L2yn–1,∀n≥1.

From (14) and (15), we have

xn+1–xn+yn+1–yn ≤1 – (1 –a)μnαn

xn–xn–1+yn–yn–1

+|μn–1–μn|

xn–1+yn–1

+PC

αn–1f(yn–1) + (1 –αn–1)T f1 n–1xn–1

+PC

αn–1g(xn–1) + (1 –αn–1)T f2

n–1yn–1

+|αn–αn–1|f(yn–1)+g(xn–1)+T f1

n–1xn–1+T f2 n–1yn–1

+ (1 –αn)

4M1 L1

s1_n–s1_n–1+4M2

L2

s2_n–s2_n–1

.

Applying Lemma4and condition (iii), we can conclude that

xn+1–xn →0 and yn+1–yn →0 asn→ ∞. (16)

Next, we show thatxn–Wn →0 whereWn=αnf(yn) + (1 –αn)T f1

nxnandyn–Vn →0

whereVn=αng(xn) + (1 –αn)T f2

nyn. Letx∈Uf1andy∈Uf2. Then we derive that

xn+1–x2 =(1 –μn)xn+μnPCWn–x2

=(1 –μn)(xn–x) +μn(PCWn–x) 2

= (1 –μn)xn–x2+μnPCWn–x2

– (1 –μn)μnxn–PCWn2

≤(1 –μn)xn–x2+μnαnf(yn) + (1 –αn)Tnf1xn–x 2

= (1 –μn)xn–x2+μnαn

f(yn) –Tnf1xn

+Tf1 nxn–x

2

– (1 –μn)μnxn–PCWn2

≤(1 –μn)xn–x2+μnTnf1xn–x 2

+ 2αn

f(yn) –T f1

nxn,αnf(yn) + (1 –αn)T f1 nxn–x

– (1 –μn)μnxn–PCWn2

≤(1 –μn)xn–x2+μnT f1 nxn–x

2

+ 2αnf(yn) –Tnf1xnαnf(yn) + (1 –αn)Tnf1xn–x

– (1 –μn)μnxn–PCWn2

≤(1 –μn)xn–x2+μnxn–x2

+ 2αnμnf(yn) –Tnf1xnαnf(yn) + (1 –αn)Tnf1xn–x

– (1 –μn)μnxn–PCWn2

=xn–x2

+ 2αnμnf(yn) –Tnf1xnαnf(yn) + (1 –αn)Tnf1xn–x

– (1 –μn)μnxn–PCWn2,

which implies that

(1 –μn)μnxn–PCWn2

≤ xn–x2–xn+1–x2

+ 2αnμnf(yn) –Tnf1xnαnf(yn) + (1 –αn)Tnf1xn–x

≤ xn–xn+1

xn–x+xn+1–x

+ 2αnμnf(yn) –Tnf1xnαnf(yn) + (1 –αn)Tnf1xn–x.

By (16), as well as conditions (i) and (ii), we get

PCWn–xn →0 asn→ ∞. (17)

From definition ofxnand applying the same method as (17), we have

PCVn–yn →0 asn→ ∞. (18)

Considering

PCWn–x2=PCWn–PCx2 ≤ Wn–x,PCWn–x

=1 2

implies that

PCWn–x ≤ Wn–x2–Wn–PCWn2. (19)

Observe that

Wn–x2=αn

f(yn) –x

+ (1 –αn)

Tf1

nxn–x2

≤αnf(yn) –x2+ (1 –αn)Tnf1xn–x2

≤αnf(yn) –x 2

+ (1 –αn)xn–x2. (20)

From (19) and (20), we obtain

xn+1–x2 =(1 –μn)xn+μnPC

αnf(yn) + (1 –αn)Tnf1xn

–x2

=(1 –μn)(xn–x) +μn(PCWn–x) 2

≤(1 –μn)xn–x2+μnPCWn–x2 ≤(1 –μn)xn–x2+μn

Wn–x2–Wn–PCWn2

≤(1 –μn)xn–x2

+μn

αnf(yn) –x 2

+ (1 –αn)xn–x2–Wn–PCWn2

,

implying that

μnWn–PCWn2≤(1 –αnμn)xn–x2–xn+1–x2+αnμnf(yn) –x 2

≤ xn–x2–xn+1–x2+αnμnf(yn) –x 2

≤ xn–xn+1

xn–x+xn+1–x

+αnμnf(yn) –x2.

Fromxn+1–xn →0 asn→ ∞and condition (i), we have

Wn–PCWn →0 asn→ ∞. (21)

From definition ofVnand applying the same argument as (21), we also obtain

Vn–PCVn →0 asn→ ∞. (22)

Since

xn–Wn=xn–PCWn+PCWn–Wn ≤ xn–PCWn+PCWn–Wn.

From (17) and (21), we have

From definition ofynand applying the same method as in (23), we also have

yn–Vn →0 asn→ ∞. (24)

Next, we show that Wn– PC(I – _L2₁∇f1)Wn → 0 as n→ ∞ and Vn –PC(I – 2

L2∇f2)Vn →0 asn→ ∞. Observe that

Wn–xn=αn

f(yn) –xn

+ (1 –αn)

Tf1 nxn–xn

,

which yields

(1 –αn)Tnf1xn–xn≤ Wn–xn+αnf(yn) –xn.

From (23) and condition (i), we have

Tf1

nxn–xn→0 asn→ ∞. (25)

Since

_Wn–Tf1

nWn=Wn–xn+xn–Tnf1xn+Tnf1xn–Tnf1Wn

≤ Wn–xn+xn–Tnf1xn+Tnf1xn–Tnf1Wn

≤ Wn–xn+xn–Tnf1xn+xn–Wn

= 2xn–Wn+Tnf1xn–xn.

From (23) and (25), we get

Tf1

nWn–Wn→0 asn→ ∞. (26)

Observe that

PC

I–λ1_n∇f1

Wn–Wn=s1nWn+

1 –s1_nTf1

nWn–Wn

=1 –s1_nTf1

nWn–Wn

≤Tf1

nWn–Wn, (27)

wheres1 n=

2–λ1nL1 4 ∈(0,

1 2).

From (27), we have

PC

I– 2

L1∇

_f1Wn–Wn

≤PC

I– 2

L1 ∇f1

Wn–PC

I–λ1_n∇f1

Wn +PC

I–λ1_n∇f1

Wn–Wn

≤

I– 2

L1∇ f1

Wn–

I–λ1_n∇f1

Wn +PC

I–λ1_n∇f1

Wn–Wn

≤

2

L1

–λ1_n∇f1(Wn)+T f1

From the boundedness of{Wn},s1n→0 (⇐⇒λ1n→L21) and (26), we conclude that

lim

n→∞ Wn–PC

I– 2

L1 ∇f1

Wn

= 0. (28)

Applying the same method as for (28), we also have

lim

n→∞ Vn–PC

I– 2

L2∇ f2

Vn

= 0. (29)

Next, we show that lim sup_n→∞f(y∗) –x∗,Wn –x∗ ≤0, where x∗ =PUf₁f(y

∗_{) and}

lim sup_n→∞g(x∗) –y∗,Vn–y∗ ≤0, wherey∗=PUf₂g(x

∗_).

Indeed, take a subsequence{Wnk}of{Wn}such that

lim sup

n→∞

fy∗–x∗,Wn–x∗

=lim sup

k→∞

fy∗–x∗,Wnk–x

∗_.

Since{xn}is bounded, without loss of generality, we may assume thatxnk xask→ ∞. From (23), we obtainWnk xask→ ∞. Assume thatx=PC(I–

2

L1∇f1)x. By

nonexpan-siveness ofPC(I–_L2₁∇f1), (28) and Opial’s property, we have

lim inf

k→∞ Wnk–x<lim inf

k→∞

Wnk–PC

I– 2

L1∇ f1

x

≤lim inf

k→∞

Wnk–PC

I– 2

L1 ∇f1

Wnk

+PC

I– 2

L1∇ f1

Wnk–PC

I– 2

L1∇ f1

x

≤lim inf

k→∞ Wnk–x.

This is a contradiction, thus we have

x∈F

PC

I– 2

L1∇

_f1=Uf1. (30)

SinceWnk xask→ ∞, due to (30) and Lemma2, we can derive that

lim sup

n→∞

fy∗–x∗,Wn–x∗

=lim sup

k→∞

fy∗–x∗,Wnk–x

∗

=fy∗–x∗,x–x∗

≤0. (31)

Similarly, take a subsequence{Vnk}of{Vn}such that

lim sup

n→∞

gx∗–y∗,Vn–y∗

=lim sup

k→∞

gx∗–y∗,Vnk–y

∗_.

obtain that

lim sup

n→∞

gx∗–y∗,Vn–y∗

≤0. (32)

Finally, we show that{xn}converges strongly tox∗, wherex∗=PUf₁f(y

∗_{) and{y}

n}

con-verges strongly toy∗, wherey∗=PUf₂g(x

∗_).

LetWn=αnf(yn) + (1 –αn)T f1

nxnandVn=αng(xn) + (1 –αn)T f2

nyn. From the definition

ofxn, we get

xn+1–x∗ 2

=(1 –μn)xn+μnPC

αnf(yn) + (1 –αn)Tnf1xn

–x∗2

=(1 –μn)

xn–x∗

+μn

PC

αnf(yn) + (1 –αn)Tnf1xn

–x∗2

= (1 –μn)xn–x∗ 2

+μnPC

αnf(yn) + (1 –αn)Tnf1xn

–x∗2

≤(1 –μn)xn–x∗ 2

+μnαnf(yn) + (1 –αn)Tnf1xn–x∗ 2

= (1 –μn)xn–x∗ 2

+μnαn

f(yn) –x∗

+ (1 –αn)

Tf1

nxn–x∗ 2

≤(1 –μn)xn–x∗ 2

+μn

(1 –αn)Tnf1xn–x∗ 2

+ 2αn

f(yn) –x∗,Wn–x∗

≤(1 –μn)xn–x∗ 2

+μn(1 –αn)xn–x∗ 2

+ 2αnμn

f(yn) –x∗,Wn–x∗

= (1 –αnμn)xn–x∗ 2

+ 2αnμn

f(yn) –x∗,Wn–x∗

= (1 –αnμn)xn–x∗2

+ 2αnμn

f(yn) –f

y∗,Wn–x∗

+fy∗–x∗,Wn–x∗

≤(1 –αnμn)xn–x∗2

+ 2αnμnf(yn) –f

y∗Wn–x∗+

fy∗–x∗,Wn–x∗

≤(1 –αnμn)xn–x∗ 2

+ 2αnμnf(yn) –f

y∗Wn–xn+1+xn+1–x∗

+ 2αnμn

fy∗–x∗,Wn–x∗

≤(1 –αnμn)xn–x∗ 2

+ 2αnμnayn–y∗Wn–xn+1+ 2αnμnayn–y∗xn+1–x∗

+ 2αnμn

fy∗–x∗,Wn–x∗

≤(1 –αnμn)xn–x∗ 2

+ 2αnμnayn–y∗Wn–xn+1+αnμnayn–y∗ 2

+xn+1–x∗ 2

+ 2αnμn

fy∗–x∗,Wn–x∗

which yields

xn+1–x∗2

≤ 1 –αnμn

1 –αnμna

xn–x∗ 2

+ 2αnμna 1 –αnμna

yn–y∗Wn–xn+1

+ αnμna 1 –αnμna

yn–y∗ 2

+ 2αnμn 1 –αnμna

fy∗–x∗,Wn–x∗

=

1 –αnμn–αnμna 1 –αnμna

xn–x∗ 2

+ 2αnμna 1 –αnμna

yn–y∗Wn–xn+1

+ αnμna 1 –αnμna

yn–y∗ 2

+ 2αnμn 1 –αnμna

fy∗–x∗,Wn–x∗

=

1 –αnμn(1 –a) 1 –αnμna

xn–x∗ 2

+ 2αnμna 1 –αnμna

yn–y∗Wn–xn+1

+ αnμna 1 –αnμna

yn–y∗ 2

+ 2αnμn 1 –αnμna

fy∗–x∗,Wn–x∗

. (33)

Similarly, as derived above, we also have

yn+1–y∗2

≤

1 –αnμn(1 –a) 1 –αnμna

yn–y∗2+

2αnμna

1 –αnμna

xn–x∗Vn–yn+1

+ αnμna 1 –αnμna

xn–x∗2+

2αnμn

1 –αnμna

gx∗–y∗,Vn–y∗

. (34)

From (33) and (34), we deduce that

xn+1–x∗ 2

+yn+1–y∗ 2

≤

1 –αnμn(1 –a) 1 –αnμna

xn–x∗ 2

+yn–y∗ 2

+ 2αnμna 1 –αnμna

yn–y∗Wn–xn+1+xn–x∗Vn–yn+1

+ αnμna 1 –αnμna

xn–x∗2+yn–y∗2

+ 2αnμn 1 –αnμna

fy∗–x∗,Wn–x∗

+gx∗–y∗,Vn–y∗

=

1 –αnμn(1 – 2a) 1 –αnμna

xn–x∗ 2

+yn–y∗ 2

+ 2αnμna 1 –αnμna

yn–y∗Wn–xn+1+xn–x∗Vn–yn+1

+ 2αnμn 1 –αnμna

fy∗–x∗,Wn–x∗

+gx∗–y∗,Vn–y∗

. (35)

By (16), (23), (24), (31), (32), condition (i) and Lemma4, we havelimn→∞(xn–x∗+yn– y∗) = 0. It implies that the sequences{xn},{yn}converge tox∗=PUf₁f(y

∗_),y∗_=P

Uf₂g(x

∗_),

Corollary 1 Let C be a nonempty closed convex subset of a real Hilbert space H.Letf :C→

Rbe a real-valued convex function and assume that∇f is 1_L-inverse strongly monotone with L> 0and U_f =∅.Let f:H→H be an a-contraction mapping with a∈(0, 1).Let the sequence{xn}be generated by x1∈C and

xn+1= (1 –μn)xn+μnPC

αnf(xn) + (1 –αn)Tnfxn

, (36)

where{μn},{αn} ⊆[0, 1],PC(I–λin∇f) =snI+ (1 –sn)T f

nand sn=2–λ₄nL,{λn} ⊂(0,2_L). As-sume that the following conditions hold:

(i) limn→∞αn= 0and ∞

n=1αn=∞,

(ii) 0 <θ≤μn≤θfor alln∈Nand for someθ,θ> 0, (iii) ∞_n=1|αn+1–αn|<∞,∞n=1|μn+1–μn|<∞.

Then{xn}converges strongly,as sn→0 (⇐⇒λn→_L2),to x∗=PUff(x

∗_).

Proof If we putf ≡g,xn=yn, in Theorem1, we obtain the desired conclusion.

4 Application

LetH1,H2be two real Hilbert spaces. LetC,Qbe nonempty closed convex subsets ofH1

andH2, respectively.

In 1994, Censor and Elfving [17] introduced thesplit feasibility problem(SFP), which is to find a pointxsuch that

x∈C and Dx∈Q,

whereD:H1→H2is a bounded linear operator.

Throughout this paper, we assume that the SFP is consistent, that is, the solution set Γ of the SFP is nonempty. Letf :H1→Rbe a continuous differentiable function. The

minimization problem

min

x∈Cf(x) :=minx∈C

1

2Ax–PQAx

2 ₍₃₇₎

is ill-posed.

Before proving Theorem2, we need the following:

Proposition 3([18]) Given x∗∈H1,the following statements are equivalent: (i) x∗solves the SFP;

(ii) PC(I–λ∇f)x∗=PC(I–λA∗(I–PQ)A)x∗=x∗;

(iii) x∗solves the variational inequality problem of findingx∗∈Csuch that

∇fx∗,x–x∗≥0, ∀x∈C, (38)

where∇f =A∗(I–PQ)AandA∗is the adjoint ofA.

Theorem 2 Let C and Q be nonempty,closed,and convex subsets of H1and H2,

mappings with af,ag∈(0, 1)and a=max{af,ag}.Let the sequences{xn},{yn}be generated by x1,y1∈C and

⎧ ⎨ ⎩

xn+1= (1 –μn)xn+μnPC(αnf(yn) + (1 –αn)Tna1xn),

yn+1= (1 –μn)yn+μnPC(αng(xn) + (1 –αn)Tna2yn),

(39)

where{μn},{αn} ⊆[0, 1],PC(I–λin(A∗i(I–PQ)Ai)) =sinI+(1–sin)T ai

n,∀i= 1, 2and sin= 2–λi_nLi

4 , {λi_n} ⊂(0, 2

Li).Assume that the following conditions hold:

(i) limn→∞αn= 0and ∞

n=1αn=∞,

(ii) 0 <θ≤μn≤θfor alln∈Nand for someθ,θ> 0, (iii) ∞_n=1|αn+1–αn|<∞,

∞

n=1|μn+1–μn|<∞.

Then{xn}and{yn}converge strongly,as sin→0 (⇐⇒λin→L2i)∀i= 1, 2,to x

∗_=P

Γ1f(y∗) withΓ1={x∈C;A1x∈Q}and y∗=PΓ2g(x∗)withΓ2={¯x∈C;A2x¯∈Q},respectively.

Proof Lettingx,y∈Cand∇fi=A∗i(I–PQ)Aifor alli= 1, 2, we have _{∇fi(x) –∇fi(y)}2

=A∗_i(I–PQ)Aix–A∗i(I–PQ)Aiy 2

≤Li(I–PQ)Aix– (I–PQ)Aiy 2

. (40)

From the property ofPC, we have (I–PQ)Aix– (I–PQ)Aiy

2

=(I–PQ)Aix– (I–PQ)Aiy, (I–PQ)Aix– (I–PQ)Aiy

=(I–PQ)Aix– (I–PQ)Aiy,Aix–Aiy

–(I–PQ)Aix– (I–PQ)Aiy,PQAix–PQAiy

=A∗_i(I–PQ)Aix–A∗i(I–PQ)Aiy,x–y

–(I–PQ)Aix– (I–PQ)Aiy,PQAix–PQAiy

=A∗_i(I–PQ)Aix–A∗i(I–PQ)Aiy,x–y

–(I–PQ)Aix,PQAix–PQAiy

+(I–PQ)Aiy,PQAix–PQAiy

≤A∗_i(I–PQ)Aix–A∗i(I–PQ)Aiy,x–y

. (41)

Substituting (41) into (40), we have

_∇fi(x) –∇fi(y) 2

≤Li

A∗_i(I–PQ)Aix–A∗i(I–PQ)Aiy,x–y

=Li

∇fi(x) –∇fi(y),x–y

.

It follows that

∇fi(x) –∇fi(y),x–y

≥ 1 Li

_∇fi(x) –∇fi(y)2.

From Proposition3and Theorem1, we can conclude that Theorem2is true.

Corollary 2 Let C and Q be nonempty,closed,and convex subsets of H1and H2, respec-tively,and let A:H1→H2 be bounded linear operator with L being the spectral radius of A∗A withΓ =∅.Let f :H→H be an a-contraction mapping with a∈(0, 1).Let the sequence{xn}be generated by x1∈C and

xn+1= (1 –μn)xn+μnPC

αnf(xn) + (1 –αn)Tna1xn

, (42)

where{μn},{αn} ⊆[0, 1],PC(I–λn(A∗(I–PQ)A)) =snI+ (1 –sn)Tna1and sn=2–λ₄nL,{λn} ⊂

(0,2_L).Assume that the following conditions hold:

(i) limn→∞αn= 0and∞n=1αn=∞,

(ii) 0 <θ≤μn≤θfor alln∈Nand for someθ,θ> 0, (iii) ∞_n=1|αn+1–αn|<∞,∞n=1|μn+1–μn|<∞.

Then {xn} converges strongly, as sn→0 (⇐⇒λn→ 2_L),to x∗ =PΓf(x∗)withΓ ={x∈

C;Ax∈Q}.

Proof If we putf ≡g,xn=ynin Theorem2, then the conclusion follows.

5 Numerical examples

Example1 LetC= [–10, 10]×[–10, 10] and let·,·:R2_×R2_{→Rbe an inner product}

defined byx, y= x·y=x1y1+x2y2, for all x = (x1,x2)∈R2 and y = (y1,y2)∈R2. For

everyi= 1, 2, letfi:C→Rbe defined byf1(x1,x2) = 2x21+x2andf2(x1,x2) = (x1– 1) +x22, ∀x1,x2∈R. Let f,g:R2→R2, defined by f(x1,x2) = (x₃1,x₃2) andg(x1,x2) = (x₄1,x₄2), be

1 2- and

1

3-contraction mappings and a=max{ 1 2,

1 3}=

1

2. Let the sequences{xn}, {yn}be

generated byx1,y1∈C. Puttingαn=₄1_nandμn=3n₇+1_n , we can rewrite (10) as follows: ⎧

⎨ ⎩

xn+1= (4n₇–1_n )xn+ (3n₇+1_n )PC(₄1_nf(yn) + (4n₄–1_n )T f1 nxn),

yn+1= (4n₇–1_n )yn+ (3n₇+1_n )PC(1₄g(xn) + (4n₄–1_n )T f2 nyn),

(43)

wherePC(x1,x2) = (max{min{x1, 10}, –10},max{min{x2, 10}, –10}) and alsoPC(I–λin∇fi) = si

nI+ (1 –sin)T fi

n andsin= 2–λi_n(16)

4 , whereλ i n= n

2

8n2₊₁ ∀i= 1, 2.

Then, sincef1(x1,x2) = 2x21+x2andf2(x1,x2) = (x1– 1) +x22, we have

∇f1(x1,x2) = (4x1, 1) and ∇f2(x1,x2) = (1, 2x2).

It is obvious that∇fiis a₁₆1-inverse strongly monotone,∀i= 1, 2.

Consider (0, –10), (–10, 0)∈[–10, 10]×[–10, 10] for which

PC

I–λ1_n∇f1

(0, –10) =P[–10,10]×[–10,10]

I– 1 16∇f1

(0, –10)

=P[–10,10]×[–10,10]

0,–161 16

=

P[–10,10](0),P[–10,10]

–161 16

=

maxmin{0, 10}, –10,max

min

–161 16 , 10

, –10

= (0, –10),

thus (0, –10)∈U_f1. Similarly,

PC

I–λ2_n∇f2

(–10, 0) =P[–10,10]×[–10,10]

I– 1 16∇f2

(–10, 0)

=P[–10,10]×[–10,10]

–161 16 , 0

=

P[–10,10]

–161 16

,P[–10,10](0)

=

max

min

–161 16 , 10

, –10

,maxmin{0, 10}, –10

= (–10, 0),

thus (–10, 0)∈U_f2.

It is clear that the sequences{αn},{μn}satisfy all the conditions of Theorem1, so we

can conclude that the sequences{xn}and{yn}converge strongly to (0, –10) and (–10, 0),

respectively. Table1 shows the values of{xn} and{yn}with x1n= –10,x2n= 10,y1n= 10, y2

n= –10, andn=N= 400.

Table 1 The values of{xn}and{yn}withx1n= –10,x2n= 10,y1n= 10,y2n= –10, andn=N= 400

n xn= (x1n,x2n) yn= (y1n,yn2)

1 (–10.0000, 10.0000) (10.0000, –10.0000) 2 (–6.0784, 8.0448) (8.1639, –7.2059) 3 (–4.2192, 7.3380) (7.5048, –5.8538) 4 (–3.0380, 6.9152) (7.1109, –4.9143) .

. .

. . .

. . .

250 (–0.0050, –7.7785) (–7.5696, –0.0076) .

. .

. . .

. . .

396 (–0.0043, –9.9937) (–9.9937, –0.0065) 397 (–0.0042, –9.9937) (–9.9937, –0.0064) 398 (–0.0042, –9.9937) (–9.9937, –0.0064) 399 (–0.0042, –9.9937) (–9.9937, –0.0064) 400 (–0.0042, –9.9937) (–9.9937, –0.0064)

Figure 1The convergence of{xn}and{yn}with

x1

Table 2 The values of{x_n}withx1

n= –10,xn2= 10, andn=N= 400

n xn= (x1n,x2n)

1 (–10.0000, 10.0000) 2 (–7.0308, 8.9972) 3 (–5.2235, 8.5685) 4 (–3.9531, 8.2876) .

. .

. . .

250 (0.0000,-6.2974) .

. .

. . .

396 (0.0000, –9.9958) 397 (0.0000, –9.9958) 398 (0.0000, –9.9958) 399 (0.0000, –9.9958) 400 (0.0000, –9.9958)

Figure 2The convergence of{xn}withx1n= –10,

x2n= 10, andn=N= 400

Remark1 If we choosef ≡gandxn=ynin Example1, we can rewrite (36) as follows:

xn+1=

4n– 1 7n

xn+

3n+ 1 7n

PC

1 4nf(xn) +

4n– 1 4n

T_nfxn

,

wherePC(x1,x2) = (max{min{x1, 10}, –10},max{min{x2, 10}, –10}) and alsoPC(I–λn∇f) = snI+ (1 –sn)T

f

nandsn=2–λn₄(16), whereλn= n 2

8n2₊₁. From Corollary1, we can conclude that

the sequence{xn}converges strongly to (0, –10). Table2shows the values of{xn} with x1_n= –10,x2_n= 10, andn=N= 400.

Conclusion

1. Theorem1guarantees the convergence of{xn}and{yn}in Example1. 2. Corollary1guarantees the convergence of{xn}in Remark1.

3. By using the concepts of an intermixed algorithm and gradient-projection algorithm (GPA), we give a new iteration for solving two constrained convex minimization problems.

Acknowledgements

This work is supported by King Mongkut’s Institute of Technology Ladkrabang.

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The two authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

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Received: 24 April 2019 Accepted: 15 October 2019 References

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